Currently in forensic science, the statistical methods for solving the identification of source problems are inherently subjective and generally ad-hoc. The formal Bayesian decision framework provides the most statistically rigorous foundation for these problems to date. However, computing a solution under this framework, which relies on a Bayes Factor, tends to be computationally intensive and highly sensitive to the subjective choice of prior distributions for the parameters. Therefore, this dissertation aims to develop statistical solutions to the forensic identification of source problems which are less subjective, but which retain the statistical rigor of the Bayesian solution.
First, this dissertation focuses on computational issues during the subjective quantification of the Bayes Factor, and on characterizing the numerical error associated with the resulting quantification. Secondly, the asymptotic properties of the Bayes Factor for a fixed set of unknown source evidence are considered as the number of control samples increases. Under the formal Bayesian paradigm, Doob's Consistency Theorem implies that a Bayesian believes in the existence of a value of evidence analogous to a true likelihood ratio in the Frequentist paradigm. Finally, two approximations to the value of evidence for the forensic identification of source problems are derived relative to the existence of a true likelihood ratio.
The first approximation is derived as a result of the Bernstein-von Mises Theorem. This Bernstein-von Mises approximation eliminates the determination of prior distributions for the parameters. Under suitable conditions, the Bernstein-von Mises approximation converges in probability to the Bayes Factor as the size of the control samples increases. However the Bernstein-von Mises approximation suffers from similar computational issues as the Bayes Factor. The second approximation is derived as a result of various theorems regarding the asymptotic properties of M-estimators. This Neyman-Pearson approximation requires no prior distributions, and is generally less computationally intractable. Under suitable conditions, the Neyman-Pearson approximation converges in probability to the true likelihood ratio as the number of control samples increases. In addition, the Neyman-Pearson approximation can replace the Bayes Factor in the forensic identification of source problems, and result in decisions that are approximately equivalent to using the Bayes Factor.
(Author abstract provided.)